Word Count : 322

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We are tasked with deriving the partial differential equation (PDE) from the given functional equation: \[ \phi \left( \frac{z}{x^{3}}, \frac{y}{z} \right) = 0 \] where \(\phi : \mathbb{R}^2 \rightarrow \mathbb{R}\) is an arbitrary function. ### Step 1: Define New Variables Let’s introduce new variables to simplify the function. Define: \[ u = \frac{z}{x^3}, \quad v = \frac{y}{z} \] Thus, the original equation becomes: \[ _______ ___ __________ __________ _____ __________ _____.
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